How to entrain a selected neuronal rhythm but not others: open-loop dithered brain stimulation for selective entrainment

Objective While brain stimulation therapies such as deep brain stimulation for Parkinson’s disease (PD) can be effective, they have yet to reach their full potential across neurological disorders. Entraining neuronal rhythms using rhythmic brain stimulation has been suggested as a new therapeutic mechanism to restore neurotypical behaviour in conditions such as chronic pain, depression, and Alzheimer’s disease. However, theoretical and experimental evidence indicate that brain stimulation can also entrain neuronal rhythms at sub- and super-harmonics, far from the stimulation frequency. Crucially, these counterintuitive effects could be harmful to patients, for example by triggering debilitating involuntary movements in PD. We therefore seek a principled approach to selectively promote rhythms close to the stimulation frequency, while avoiding potential harmful effects by preventing entrainment at sub- and super-harmonics. Approach Our open-loop approach to selective entrainment, dithered stimulation, consists in adding white noise to the stimulation period. Main results We theoretically establish the ability of dithered stimulation to selectively entrain a given brain rhythm, and verify its efficacy in simulations of uncoupled neural oscillators, and networks of coupled neural oscillators. Furthermore, we show that dithered stimulation can be implemented in neurostimulators with limited capabilities by toggling within a finite set of stimulation frequencies. Significance Likely implementable across a variety of existing brain stimulation devices, dithering-based selective entrainment has potential to enable new brain stimulation therapies, as well as new neuroscientific research exploiting its ability to modulate higher-order entrainment.


B Simulating neural oscillators coupled through Hodgkin-Huxley electrotonic coupling
To verify that dithered stimulation is still effective in populations of neural oscillators coupled through electrotonic coupling derived from the Hodgkin-Huxley model [51], we also considered a modified version of equation (11) in the main text. In this modified model, the time evolution of the phase φ k of the k th oscillator is described by where G e denotes the electrotonic coupling function derived from the Hodgkin-Huxley model in [51] (see Fig S.9). To get a large 2:1 synchronisation region in the absence of dithering for meaningful testing, we chose κ = 300, ξ = 7.9, and sampled the ω k 's from a Lorentzian distribution centered on the frequency considered (f 0 /[2π]) and of width 20 Hz.

C Frequencies used in toggling analysis
The frequencies used in the first two rows of   , 108.33, 118.18, 130, 144.44, 162.50, 185.71} Hz.
The frequencies used in the last two rows of Fig 6 in the main text, and Figs S.6 and S.7 are {100, 104, 108.33, 113.04, 118.18, 123.81, 130, 136.84, 144.44, 152.94, 162.50, 173.33, 185.71} Hz. Dithering level increases from top to bottom, and theoretical tongue boundaries (equations (6) and (9)) are shown by black dashed lines. The frequency corresponding to the mean stimulation period is indicated by a red dashed line. An animation with finer steps in dithering levels is provided as supplementary material (jneacbc4asupp3.mp4, see Video S.2 for caption). For each natural frequency, stimulation amplitude, and dithering value, the rotation number is averaged over 10 repeats, with 10 4 stimulation pulses per repeat. Higher dithering levels, resulting in only the 1:1 tongue being stable, are shown in Fig 3A in the main text.   Here, PLV p:1 detects 1:1 and 2:1 entrainment. In all panels, for each natural frequency, stimulation amplitude, and dithering value, the rotation number or mean instantaneous frequency is averaged over 5 repeats, with 400 stimulation pulses per repeat. The frequency corresponding to the mean stimulation period is indicated by a red dashed line. For ζ = 0.15, 2:1 entrainment has disappeared while 1:1 entrainment is still supported (compare B2 to B1). See Section A for more details on the PLV. The modified model used in this figure is defined in Section B.

Figure S.5: The efficacy of dithered stimulation is confirmed by PLV analysis in populations of coupled neural oscillators.
Dithering level (ζ) increases from the top row to the bottom row. A: Frequency locking regions in the natural frequency/stimulation amplitude space. Only regions of frequency-locking (determined as presented in Section 4.2 in the main text), are shown in color. The color scale represents the rotation number. B: PLV p:1 (color scale) in the natural frequency/stimulation amplitude space. Here, PLV p:1 detects 1:1 and 2:1 entrainment. C: PLV p:2 (color scale) in the natural frequency/stimulation amplitude space. Here, PLV p:2 detects 1:2 and 3:2 entrainment, but also 1:1 and 2:1 entrainment. D: PLV (2p − 1):2 (color scale) in the natural frequency/stimulation amplitude space, obtained as PLV p:2 -PLV p:1. Here, PLV (2p − 1):2 detects 1:2 and 3:2 entrainment. In all panels, for each natural frequency, stimulation amplitude, and dithering value, the rotation number or mean instantaneous frequency is averaged over 5 repeats, with 400 stimulation pulses per repeat. The frequency corresponding to the mean stimulation period is indicated by a red dashed line. For ζ = 0.15, 2:1 entrainment has disappeared while 1:1 entrainment is still supported (compare B4 to B1). 1:2 and 3:2 entrainment have also disappeared (compare D4 to D1). See Section A for more details on the PLV. Here, PLV p:1 detects 1:1 and 2:1 entrainment. C: PLV p:2 (color scale) in the natural frequency/stimulation amplitude space. Here, PLV p:2 detects 1:2 and 3:2 entrainment, but also 1:1 and 2:1 entrainment. D: PLV (2p − 1):2 (color scale) in the natural frequency/stimulation amplitude space, obtained as PLV p:2 -PLV p:1. Here, PLV (2p − 1):2 detects 1:2 and 3:2 entrainment. In all panels, for each natural frequency, stimulation amplitude, and dithering value, the rotation number or mean instantaneous frequency is averaged over 5 repeats, with 400 stimulation pulses per repeat. The frequency corresponding to the mean stimulation period is indicated by a red dashed line. The stimulation frequencies used are plotted as red diamonds on the left of the figure (numerical values are given in Section C). See Section A for more details on the PLV. Here, PLV p:2 detects 1:2 and 3:2 entrainment, but also 1:1 and 2:1 entrainment. D: PLV (2p − 1):2 (color scale) in the natural frequency/stimulation amplitude space, obtained as PLV p:2 -PLV p:1.
Here, PLV (2p − 1):2 detects 1:2 and 3:2 entrainment. In all panels, for each natural frequency, stimulation amplitude, and dithering value, the rotation number or mean instantaneous frequency is averaged over 5 repeats, with 400 stimulation pulses per repeat. The frequency corresponding to the mean stimulation period is indicated by a red dashed line. The stimulation frequencies used are plotted as red diamonds on the left of the figure (numerical values are given in Section C). See Section A for more details on the PLV.